Indeed, setting \(\Phi _1(t,x)=e^{i\lambda _1t}u(x)\)  and \(\Phi _2(t,x)=e^{i\lambda _2t}v(x)\) , if \((u,v)\)  is a nontrivial solution of (REF ), then \((\Phi _1,\Phi _2)\)  is a solitary wave solution of (REF ). Physically, system (REF ) has the nature of conservation of mass, i.e., the \(L^2\) -norms \(\Vert \Phi _1(t,\cdot )\Vert ^2_2\)  and \(\Vert \Phi _2(t,\cdot )\Vert ^2_2\)  of solutions are independent of \(t\in \mathbb {R}\) . Problems with prescribed masses
(the former constraint) appear in nonlinear optics, where the mass represents the power supply, and in the theory of Bose-Einstein condensates, where it represents the total number of particles (see [1]}, [2]}, [3]}, [4]}, [5]}). In particular, the positive sign of \(\mu _1,\mu _2, \beta \)  stays for attractive interaction, while the negative sign stays for repulsive interaction.
